"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." – Paul Halmos

Archive for September, 2009

Theorem: The set of finite sequences of elements of a countable set is countable.

I like this result because it specializes to several other basic countability results: for example, it implies that countable unions, finite products, and the set of finite subsets of countable sets are countable. I know several proofs of this result and I am honestly curious which ones people prefer.

I have at least four planned posts left in the series on symmetric functions, but unfortunately I’ll be very busy for the next two weeks. In the meantime, here are some thoughts on the primes.

Euclid’s proof of the infinitude of the primes is often held up as a shining example of mathematical proof. (Whether this reputation is deserved is a matter of opinion.) Euler’s proof via the zeta function is also classic. An even showier proof by Furstenburg is phrased in the language of topology. Today I’d like to share my personal favorite proof and discuss one of its possible consequences.